Uniform norm for all functions

Question

Uniform norm on the space of continuous functions from [0,T] on $\Bbb R$ is

$\|f\|=\max_{t\in[0,T]} |f(x)|$

I am wondering why can't we extend this norm to all functions including functions that are not continuous? What properties of a norm do not for a discontinuous function?

Answer 1

Non continuous functions are not necessarily bounded, and so the maximum need not exist. You can however replace max by supremum and extend to the space of all bounded functions.